Strange Equations Quiz Puzzle Answer

We saw a very strange equation a long time ago written on a bar mat in the Jolly Quiz Master pub. As we said, it was a long time ago, but we kept the bar mat and found it again recently, and thought we would share it with you.

$$ \frac{1}{4} X \frac{8}{5} = \frac{18}{45} $$

To be completely correct you should cancel out the nines to leave two fifths, but the equation actually works even though the method used to produce it is decidedly dodgy.

This set us wondering whether there are any other equations where this incorrect method produces correct equations. Using single-digit, non-zero numbers (that is, 1 through to 9) on the left-hand side of the equation, how many correct equations can you find that follow this method?

If the four digits are a, b, c and d, we are trying to solve the equation

$$ \frac{a}{b} X \frac{c}{d} = \frac{10a + c}{10b + d} $$


$$ ac(10b + d) = bd(10a + c) $$

Ignoring the trivial cases when a = b and c = d there are seven solutions, which can be inverted to make another seven, giving 14 in all. The first seven are:

$$ \frac{1}{2} X \frac{5}{4} = \frac{15}{24} $$

$$ \frac{1}{4} X \frac{8}{5} = \frac{18}{45} $$

$$ \frac{1}{6} X \frac{4}{3} = \frac{14}{63} $$

$$ \frac{1}{6} X \frac{6}{4} = \frac{16}{64} $$

$$ \frac{1}{9} X \frac{9}{5} = \frac{19}{95} $$

$$ \frac{2}{6} X \frac{6}{5} = \frac{26}{65} $$

$$ \frac{4}{9} X \frac{9}{8} = \frac{49}{98} $$

We'll let you invert them if you want!


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